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Concept
Liar paradox
Summary
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.
The Epimenides paradox (c. 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated that "All Cretans are liars." However, Epimenides' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie (alternatively, it can be taken as merely a statement that all Cretans tell lies, not that they tell only lies).
The paradox's name translates as pseudómenos lógos (ψευδόμενος λόγος) in Ancient Greek. One version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus, who lived in the 4th century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"
The paradox was once discussed by St. Jerome in a sermon:
"I said in my alarm, Every man is a liar!" Is David telling the truth or is he lying? If it is true that every man is a liar, and David's statement, "Every man is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement that "Every man is a liar", consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction.
Official source
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